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Chapter 7 Geodesics on Riemannian Manifolds

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Chapter 7 Geodesics on Riemannian Manifolds Chapter 7 Geodesics on Riemannian Manifolds 7.1 Geodesics, Local Existence and Uniqueness If (M,g)isaRiemannianmanifold,thentheconceptof length makes sense for any piecewise smooth (in fact, C1) curve on M. Then, it possible to define the structure of a metric space on M,whered(p, q)isthegreatestlowerboundofthe length of all curves joining p and q. Curves on M which locally yield the shortest distance between two points are of great interest. These curves called geodesics play an important role and the goal of this chapter is to study some of their properties. 489 490 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS Given any p M,foreveryv TpM,the(Riemannian) norm of v,denoted∈ v ,isdefinedby∈ " " v = g (v,v). " " p ! The Riemannian inner product, gp(u, v), of two tangent vectors, u, v TpM,willalsobedenotedby u, v p,or simply u, v .∈ # $ # $ Definition 7.1.1 Given any Riemannian manifold, M, a smooth parametric curve (for short, curve)onM is amap,γ: I M,whereI is some open interval of R. For a closed→ interval, [a, b] R,amapγ:[a, b] M is a smooth curve from p =⊆γ(a) to q = γ(b) iff→γ can be extended to a smooth curve γ:(a ", b + ") M, for some ">0. Given any two points,− p, q →M,a ∈ continuous map, γ:[a, b] M,isa" piecewise smooth curve from p to q iff → (1) There is a sequence a = t0 <t1 < <tk 1 <t = b of numbers, t R,sothateachmap,··· − k i ∈ γi = γ ! [ti,ti+1], called a curve segment is a smooth curve, for i =0,...,k 1. − (2) γ(a)=p and γ(b)=q. 7.1. GEODESICS, LOCAL EXISTENCE AND UNIQUENESS 491 The set of all piecewise smooth curves from p to q is denoted by Ω(M; p, q)orbrieflybyΩ(p, q)(orevenby Ω, when p and q are understood). The set Ω(M; p, q)isanimportantobjectsometimes called the path space of M (from p to q). Unfortunately it is an infinite-dimensional manifold, which makes it hard to investigate its properties. Observe that at any junction point, γi 1(ti)=γi(ti), there may be a jump in the velocity vector− of γ. We let γ(((ti)+)=γi((ti)andγ(((ti) )=γi( 1(ti). − − 492 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS Given any curve, γ Ω(M; p, q), the length, L(γ), of γ is defined by ∈ k 1 − ti+1 L(γ)= γ((t) dt " " i=0 $ti #k 1 − ti+1 = g(γ((t),γ((t)) dt. i=0 ti # $ % It is easy to see that L(γ)isunchangedbyamonotone reparametrization (that is, a map h:[a, b] [c, d], whose → derivative, h(,hasaconstantsign). Let us now assume that our Riemannian manifold, (M,g), is equipped with the Levi-Civita connection and thus, for D every curve, γ,onM,letdt be the associated covariant derivative along γ,alsodenoted ∇γ( 7.1. GEODESICS, LOCAL EXISTENCE AND UNIQUENESS 493 Definition 7.1.2 Let (M,g)beaRiemannianmani- fold. A curve, γ: I M,(whereI R is any interval) → ⊆ is a geodesic iff γ((t)isparallelalongγ,thatis,iff Dγ( = γ( =0. dt ∇γ( If M was embedded in Rd,ageodesicwouldbeacurve, Dγ( γ,suchthattheaccelerationvector,γ(( = dt ,isnormal to Tγ(t)M. By Proposition 6.4.6, γ((t) = g(γ (t),γ(t)) is con- " " ( ( stant, say γ((t) = c. " " % 494 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS If we define the arc-length function, s(t), relative to a, where a is any chosen point in I,by t s(t)= g(γ (t),γ(t)) dt = c(t a),tI, ( ( − ∈ $a we conclude that% for a geodesic, γ(t), the parameter, t,is an affine function of the arc-length. The geodesics in Rn are the straight lines parametrized by constant velocity. The geodesics of the 2-sphere are the great circles, parametrized by arc-length. The geodesics of the Poincar´ehalf-plane are the lines x = a and the half-circles centered on the x-axis. The geodesics of an ellipsoid are quite fascinating. They can be completely characterized and they are parametrized by elliptic functions (see Hilbert and Cohn-Vossen [?], Chapter 4, Section and Berger and Gostiaux [?], Section 10.4.9.5). 7.1. GEODESICS, LOCAL EXISTENCE AND UNIQUENESS 495 If M is a submanifold of Rn,geodesicsarecurveswhose acceleration vector, γ(( =(Dγ()/dt is normal to M (that is, for every p M, γ(( is normal to T M). ∈ p In a local chart, (U,ϕ), since a geodesic is characterized by the fact that its velocity vector field, γ((t), along γ is parallel, by Proposition 6.3.4, it is the solution of the following system of second-order ODE’s in the unknowns, uk: d2u du du k + Γk i j =0,k=1,...,n, dt2 ij dt dt ij # with u = pr ϕ γ (n =dim(M)). i i ◦ ◦ 496 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS The standard existence and uniqueness results for ODE’s can be used to prove the following proposition: Proposition 7.1.3 Let (M,g) be a Riemannian man- ifold. For every point, p M,andeverytangentvec- ∈ tor, v TpM,thereissomeinterval,( η,η),anda unique∈ geodesic, − γ :( η,η) M, v − → satisfying the conditions γv(0) = p, γv( (0) = v. The following proposition is used to prove that every geodesic is contained in a unique maximal geodesic (i.e, with largest possible domain): 7.1. GEODESICS, LOCAL EXISTENCE AND UNIQUENESS 497 Proposition 7.1.4 For any two geodesics, γ : I M and γ : I M,ifγ (a)=γ (a) and 1 1 → 2 2 → 1 2 γ1( (a)=γ2( (a),forsomea I1 I2,thenγ1 = γ2 on I I . ∈ ∩ 1 ∩ 2 Propositions 7.1.3 and 7.1.4 imply that for every p M and every v T M,thereisauniquegeodesic,denoted∈ ∈ p γv,suchthatγ(0) = p, γ((0) = v,andthedomainofγ is the largest possible, that is, cannot be extended. We call γv a maximal geodesic (with initial conditions γv(0) = p and γv( (0) = v). Observe that the system of differential equations satisfied by geodesics has the following homogeneity property: If t γ(t)isasolutionoftheabovesystem,thenforevery constant,,→ c,thecurvet γ(ct)isalsoasolutionofthe system. ,→ 498 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS We can use this fact together with standard existence and uniqueness results for ODE’s to prove the proposition below. Proposition 7.1.5 Let (M,g) be a Riemannian man- ifold. For every point, p0 M,thereisanopensub- set, U M,withp U∈,andsome">0,sothat: ⊆ 0 ∈ For every p U and every tangent vector, v TpM, with v <"∈,thereisauniquegeodesic, ∈ " " γ :( 2, 2) M, v − → satisfying the conditions γv(0) = p, γv( (0) = v. If γ :( η,η) M is a geodesic with initial conditions v − → γv(0) = p and γv( (0) = v =0,foranyconstant,c =0,the curve, t γ (ct), is a geodesic- defined on ( η-/c, η/c) ,→ v − (or (η/c, η/c)ifc<0) such that γ((0) = cv.Thus, − γ (ct)=γ (t),ct( η,η). v cv ∈ − This fact will be used in the next section. 7.2. THE EXPONENTIAL MAP 499 7.2 The Exponential Map The idea behind the exponential map is to parametrize aRiemannianmanifold,M,locallynearanyp M ∈ in terms of a map from the tangent space TpM to the manifold, this map being defined in terms of geodesics. Definition 7.2.1 Let (M,g)beaRiemannianmani- fold. For every p M,let (p)(orsimply, )bethe ∈ D D open subset of TpM given by (p)= v T M γ (1) is defined , D { ∈ p | v } where γv is the unique maximal geodesic with initial con- ditions γv(0) = p and γv( (0) = v.Theexponential map is the map, exp : (p) M,givenby p D → expp(v)=γv(1). It is easy to see that (p)isstar-shaped,whichmeans that if w (p), thenD the line segment tw 0 t 1 is contained∈D in (p). { | ≤ ≤ } D 500 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS In view of the remark made at the end of the previous section, the curve t exp (tv),tv(p) ,→ p ∈D is the geodesic, γv,throughp such that γv( (0) = v.Such geodesics are called radial geodesics .Thepoint,expp(tv), is obtained by running along the geodesic, γv,anarc length equal to t v ,startingfromp. " " In general, (p)isapropersubsetofT M. D p Definition 7.2.2 ARiemannianmanifold,(M,g), is geodesically complete iff (p)=T M,forallp M, D p ∈ that is, iffthe exponential, expp(v), is defined for all p M and for all v T M. ∈ ∈ p Equivalently, (M,g)isgeodesicallycompleteiffevery geodesic can be extended indefinitely. Geodesically complete manifolds have nice properties, some of which will be investigated later. Observe that d(expp)0 =idTpM . 7.2. THE EXPONENTIAL MAP 501 It follows from the inverse function theorem that expp is adiffeomorphismfromsomeopenballinTpM centered at 0 to M. The following slightly stronger proposition can be shown: Proposition 7.2.3 Let (M,g) be a Riemannian man- ifold. For every point, p M,thereisanopensubset, W M,withp W and∈ a number ">0,sothat ⊆ ∈ (1) Any two points q1,q2 of W are joined by a unique geodesic of length <". (2) This geodesic depends smoothly upon q1 and q2, that is, if t expq1(tv) is the geodesic joining q and q (0 ,→ t 1), then v T M depends 1 2 ≤ ≤ ∈ q1 smoothly on (q1,q2). (3) For every q W ,themapexp is a diffeomor- ∈ q phism from the open ball, B(0,") TqM,toits image, U =exp(B(0,")) M,with⊆ W U and q q ⊆ ⊆ q Uq open.
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